Two-Steps Versus One-Step Solidification Pathways of Binary Metallic Nanodroplets

The solidification of AgCo, AgNi, and AgCu nanodroplets is studied by molecular dynamics simulations in the size range of 2–8 nm. All these systems tend to phase separate in the bulk solid with surface segregation of Ag. Despite these similarities, the simulations reveal clear differences in the solidification pathways. AgCo and AgNi already separate in the liquid phase, and they solidify in configurations close to equilibrium. They can show a two-step solidification process in which Co-/Ni-rich parts solidify at higher temperatures than the Ag-rich part. AgCu does not separate in the liquid and solidifies in one step, thereby remaining in a kinetically trapped state down to room temperature. The solidification mechanisms and the size dependence of the solidification temperatures are analyzed, finding qualitatively different behaviors in AgCo/AgNi compared to AgCu. These differences are rationalized by an analytical model.

1: Final geometric structures for different sizes and cooling rates. The nanoparticle structures are classified as face centered cubic (fcc), decahedral (Dh), icosahedral (Ih) and polyicosahedral (polyIh). In all cases 10 independent simulations are run. All structures are defective, for example fcc structures very often present twin planes and/or hcp parts. Some decahedral structures are indeed bi-or tri-decahedra.  8  0  1000  AgCu  10 1  1  8  0  2000  AgCo  1 2  4  4  0  2000  AgCo  10 2  3  5  0  2000  AgNi  1 4  0  6  0  2000  AgNi  10 1  1  8  0  2000  AgCu  1 2  0  8  0  2000  AgCu  10 1  1  8  0  4000  AgCo  1 5  3  2  0  4000 AgNi 1  Kinetic trapping down to small sizes  Fig. S1) indicating that the chemical ordering obtained in the freezing simulations is not optimal. This is especially true for AgCu, in which the energy gain is by far the largest. Therefore, also for size 250, kinetic trapping is stronger in AgCu. To verify whether kinetic trapping persists on longer times scales, we performed 10 simulations of AgCu at the slower cooling rate of 0.1 K/ns, finding a somewhat smaller energy gain after optimization of chemical ordering (∆E = −1.18 eV), which is however larger than the ∆E values of AgCo and AgNi found at the faster cooling rate.

Solid nucleus in AgNi
In Fig. S2 we show the complete icosahedral nucleus which starts the solidification process in a freezing simulation of Ag 375 Ni 125 with cooling rate 1 K/ns. The nucleus is the same as

Examples of caloric curves
In Figure S3 we report the caloric curves obtained in the 10 independent simulations of Ag 3000 Cu 1000 with a cooling rate of 0.1 K/ns. The simulations started at T = 1100 K and were continued down to 400 K, but in the figure we show only the temperature range in which the jumps occur. In each simulation, the jump is quite sharp, but it changes its position from simulation to simulation.
Jumps are generally sharp for all cooling rates, with the exception of the solidification of small Ni-rich parts in which the solidifying part may oscillate between liquid and solid states in temperature intervals of the order of 10-15 K.
The solidification temperatures reported in the main text ( Fig. 6(a,b)) have been cal-culated as averages on 10 simulations. The error bars on these averages correspond to one standard deviation.

Alternative models for T sol
We start from Eq. (1) of the main text, 4 which is repeated here If we assume that in general N c = cf (N ), where f is a generic function, the expression for . (S2) If f (N ) has a power-law behaviour, f (N ) = cN δ , the functional form of T sol becomes where α ′ = α/δ and γ ′ = γ/δ, so that this type of functional dependence of T sol does not discriminate for example between droplets in which the density of nucleation centers scales as the volume and droplets in which nucleation centers scale proportionally to the surface area (∼ N 2/3 ).
An alternative approximation uses for T sol the same functional of dependence as in the Pawlow's law 5 for equilibrium melting, i.e.
Sheng et al. 6 used this kind of size dependence also for solidification, which is a kinetic phenomenon, in the case of heterogeneous nucleation of single-component nanoparticles.
The justification of the use of this formula for solidification relies on several assumptions: • Classical nucleation theory 4 is used for estimating the formation rate of a critical solid nucleus.
• In the expression of the nucleation rate (Eq. (S1)) the only size dependence is assumed to derive from the term corresponding to the free-energy difference per unit volume at equilibrium between liquid and solid phases, while N c is assumed not to depend on N .
The size dependence of the free energy difference is due to the interface free energy between the droplet and its environment. Parameters of the fits of Figure 6(b) of the main text The parameters α and γ of Eq. (7) of the main text i.e.
T sol = α/(γ − ln(N )) (S5) used to fit the data of Fig. 6(b) of the main text are given in Table S2. For each cooling rate, we fit both parameters independently.
We note that, as derived in the main text, α = aT inst − Q, which should not depend on the transition rate. From the results in Table S2, it turns out that the α values for the cooling rates of 0.1 and 1 K/ns are quite close, since their difference is compatible with zero within 1.1 standard deviations. On the contrary, the α value for the cooling rate of 10 K/ns is significantly larger, thus being not compatible with the other values. This indicates that the linear expression of ∆G in terms of (T − T inst ) is able to to capture the main characteristics Comparison of the models with the simulation data for

AgCu
In Figure S4 we compare the best fits of Eq. (S5) (solid lines) and of Eq. (S4) (dash-dotted lines) with the simulation data for AgCu (symbols). We already verified that Eq. (S5) very nicely fits the data for all cooling rates, with quantitative agreement. On the contrary, Eq.
(S4) fits well only the data for the cooling rate of 10 K/ns, while the fits for 1 K/ns and 0.1 K/ns are poor from the quantitative point of view (see the p-values of the χ-square test in Table S3).

Solidification temperatures of AgCo and AgNi
In Figures S5 and S6 we report the solidification temperatures for the Ag-rich parts and the Co/Ni rich parts, respectively. The simulation data are compared to the best fits of Eq. (S5) (solid lines) and of Eq. (S4) (dash-dotted lines). In all cases, no quantitative agreement is obtained, but Eq. (S4) in much better qualitative agreement with the simulation data than Eq. (S5).  with the simulation data for the solidification temperature T sol of the Co-rich and Ni-rich parts for cooling rate of 1 K/ns. The data are reported as a function of the number N Co/N i of Co or Ni atoms. Blue lines and symbols correspond to AgCo whereas green lines and symbols correspond to AgNi. The error bars on the symbols correspond to one standard deviation of the average T sol over 10 independent simulations. Temperatures are in K. We note that for AgNi, the Ni-rich part solidifies at higher temperatures than the Ag-rich part for all sizes, whereas in AgCo separate solidification (i.e. the two-step pathway) occurs only for N Co ≥ 500 (N ≥ 2000).